Generalized Capacity Planning for the Hospital-Residents Problem

TL;DR

This work studies a generalized capacity planning problem for the Hospital-Residents setting by introducing costs to augment program quotas, with the aim of achieving an $ ext{A}$-perfect stable matching. It defines two optimization goals, MinSum (minimize total augmentation cost) and MinMax (minimize the maximum program-wise augmentation cost), and establishes a sharp complexity contrast: MinMax is solvable in polynomial time, while MinSum is NP-hard and hard to approximate in general. The authors provide a suite of approximation algorithms for MinSum, including a $|\mathcal{P}|$-approximation, an $\ell_p$-approximation, and, for the two-cost MinSumC case, an $\ell_a$-approximation via LP-based methods; they also prove constant-factor and UGC-based inapproximability results. Hardness results are derived through reductions from Set Cover and Vertex Cover, and the work further clarifies the relationship between envy-free and stable matchings in the HR context, offering practical insights for capacity planning under costs. Overall, the paper advances the theoretical foundation for cost-informed quota augmentation in two-sided matching and suggests directions for tighter approximation guarantees and broader algorithmic extensions.

Abstract

The Hospital Residents setting models important problems like school choice, assignment of undergraduate students to degree programs, among many others. In this setting, fixed quotas are associated with the programs that limit the number of agents that can be assigned to them. Motivated by scenarios where all agents must be matched, we propose and study a generalized capacity planning problem, which allows cost-controlled flexibility with respect to quotas. Our setting is an extension of the Hospital Resident setting where programs have the usual quota as well as an associated cost, indicating the cost of matching an agent beyond the initial quotas. We seek to compute a matching that matches all agents and is optimal with respect to preferences, and minimizes either a local or a global objective on cost. We show that there is a sharp contrast -- minimizing the local objective is polynomial-time solvable, whereas minimizing the global objective is NP-hard. On the positive side, we present approximation algorithms for the global objective in the general case and a particular hard case. We achieve the approximation guarantee for the special hard case via a linear programming based algorithm. We strengthen the NP-hardness by showing a matching lower bound to our algorithmic result.

Figures (9)

• Figure 1: An illustrative example for MinSum and MinMax.
• Figure 2: Promotion of agents who envy
• Figure 3: A schematic depicting the various types of programs. Blue edges denote edges in initial matching $M_I$ while red edges indicate the edges between an agent and its least-cost program. The graph contains edges from the matching $M_L$.
• Figure 4: Linear Program and its dual for the $\textsc{MinSumC}\xspace$ problem
• Figure 5: Let $(a',p,a)$ be a valid triplet and $p' \succ_{a'} p$. The edges shown in the figure are those whose dual constraint contains the variable $z_{a',p,a}$ in either positive or negative form.

Theorems & Definitions (50)

• definition 1: Stable Matching
• definition 2: Envy-Freeness
• theorem 1
• theorem 2
• theorem 3
• theorem 4
• theorem 5
• theorem 6
• lemma 1
• proof